FA-93511 / Solar tracker geometry / Open access
Heliostat mirror normal aiming: sun vector components · case 01
Mirrors aim as if the sun were mirrored about the north-east diagonal.
ROOT CAUSE
The sun vector is built with azimuth measured from east.
VERIFIED REPAIR
Use east=cos(el)sin(az), north=cos(el)cos(az).
Unsuccessful approach: Dropping the cos(el) factor makes the sun vector non-unit.
Case contract
mirror and receiver are [east, north, up] positions in metres. The mirror normal bisects the unit sun vector (east=cos(el)sin(az), north=cos(el)cos(az), up=sin(el)) and the unit vector from the mirror to the receiver. Return [normal azimuth 0..360 clockwise from north, normal elevation], rounded to 3; None when sun_el <= 0; 'invalid' when receiver equals mirror.
Why this case matters
Single-axis and dual-axis solar trackers turn a sun direction into actuator commands; a sign, frame or limit mistake points a whole plant away from the sun or into a mechanical stop.
1 / The failure
Exit 1"""Failure Map reference implementation. Python standard library only."""
import json
import math
N = 1
observations = []
def solve(sun_az, sun_el, mirror, receiver):
if sun_el <= 0:
return None
dx = [receiver[i] - mirror[i] for i in range(3)]
d = math.sqrt(sum(v * v for v in dx))
if d == 0:
return 'invalid'
t = [v / d for v in dx]
az = math.radians(sun_az)
el = math.radians(sun_el)
s = [math.cos(el) * math.cos(az), math.cos(el) * math.sin(az), math.sin(el)]
h = [s[i] + t[i] for i in range(3)]
m = math.sqrt(sum(v * v for v in h))
h = [v / m for v in h]
n_az = math.degrees(math.atan2(h[0], h[1])) % 360
n_el = math.degrees(math.asin(h[2]))
return [round(n_az, 3), round(n_el, 3)]
def check(label, actual, expected):
observations.append({"check": label, "actual": actual, "expected": expected, "passed": actual == expected})
fixtures = [[['boundary: receiver at mirror', [180, 40, [0, 0, 0], [0, 0, 0]], 'invalid'],
['normal: receiver due north on tower', [180, 40, [0, -40, 0], [0, 0, 40]], [180.0, 87.5]],
['boundary: night', [90, 0, [10, 10, 0], [0, 0, 50]], None],
['regression: sun vector components', [270, 60, [6, -1, 1], [0, 5, 0]], [300.29, 28.278]],
['regression: sun vector components (partial repair)', [270, 80, [-16, -19, 1], [0, 0, 0]],
[31.588, 46.473]],
['control 1', [90, -3, [-12, 43, 1], [0, 0, 0]], None],
['control 2', [200, 60, [-10, 12, 0], [0, 0, 60]], [180.839, 70.11]]],
[['boundary: receiver at mirror', [180, 40, [0, 0, 0], [0, 0, 0]], 'invalid'],
['normal: receiver due north on tower', [180, 40, [0, -40, 0], [0, 0, 40]], [180.0, 87.5]],
['boundary: night', [90, 0, [10, 10, 0], [0, 0, 50]], None],
['regression: sun vector components', [315, 25, [47, -28, 0], [0, 0, 60]], [308.943, 36.527]],
['regression: sun vector components (partial repair)', [90, 25, [-19, 40, 2], [-10, 5, 60]],
[116.262, 47.684]],
['control 1', [135, 60, [31, 24, 2], [10, 0, 60]], [177.1, 67.63]],
['control 2', [225, -3, [-22, -29, 0], [0, 5, 30]], None]],
[['boundary: receiver at mirror', [180, 40, [0, 0, 0], [0, 0, 0]], 'invalid'],
['normal: receiver due north on tower', [180, 40, [0, -40, 0], [0, 0, 40]], [180.0, 87.5]],
['boundary: night', [90, 0, [10, 10, 0], [0, 0, 50]], None],
['regression: sun vector components', [60, 40, [7, -5, 0], [0, 0, 60]], [49.649, 66.233]],
['regression: sun vector components (partial repair)', [315, 10, [-37, 26, 0], [0, -5, 60]],
[323.691, 69.061]],
['control 1', [90, 10, [-6, -40, 0], [-10, 0, 30]], [48.617, 32.61]],
['control 2', [30, 10, [-44, 5, 1], [-10, 0, 0]], [64.473, 5.033]]],
[['boundary: receiver at mirror', [180, 40, [0, 0, 0], [0, 0, 0]], 'invalid'],
['normal: receiver due north on tower', [180, 40, [0, -40, 0], [0, 0, 40]], [180.0, 87.5]],
['boundary: night', [90, 0, [10, 10, 0], [0, 0, 50]], None],
['regression: sun vector components', [315, 60, [-28, -33, 2], [-10, 5, 60]], [353.399, 62.028]],
['regression: sun vector components (partial repair)', [200, 80, [9, 42, 3], [-10, 0, 30]],
[203.614, 55.19]],
['control 1', [150, 40, [16, -12, 0], [0, 0, 0]], [261.353, 56.729]],
['control 2', [90, 0, [46, -4, 0], [0, -5, 0]], None]],
[['boundary: receiver at mirror', [180, 40, [0, 0, 0], [0, 0, 0]], 'invalid'],
['normal: receiver due north on tower', [180, 40, [0, -40, 0], [0, 0, 40]], [180.0, 87.5]],
['boundary: night', [90, 0, [10, 10, 0], [0, 0, 50]], None],
['regression: sun vector components', [200, 10, [41, -47, 2], [0, 0, 30]], [255.744, 31.113]],
['regression: sun vector components (partial repair)', [225, 40, [-5, 13, 2], [-10, -5, 30]],
[212.668, 49.07]],
['control 1', [250, -3, [-49, 36, 3], [-10, 5, 60]], None],
['control 2', [300, 80, [-27, -17, 1], [-10, 0, 60]], [18.225, 78.974]]]]
for label, args, expected in fixtures[N-1]:
check(label, solve(*args), expected)
print(json.dumps({"observations": observations, "passed": all(x["passed"] for x in observations)}, ensure_ascii=False))
raise SystemExit(0 if all(x["passed"] for x in observations) else 1)
| Boundary fixture | Actual | Expected | Outcome |
|---|---|---|---|
| boundary: receiver at mirror | invalid | invalid | Passed |
| normal: receiver due north on tower | [312.709, 52.321] | [180.0, 87.5] | Failed |
| boundary: night | None | None | Passed |
| regression: sun vector components | [286.066, 45.704] | [300.29, 28.278] | Failed |
| regression: sun vector components (partial repair) | [47.457, 47.237] | [31.588, 46.473] | Failed |
| control 1 | None | None | Passed |
| control 2 | [220.244, 75.402] | [180.839, 70.11] | Failed |
SHA-256 / 9280dcdcf47dc6b27209a56366d90fe9c2f1e5adf116d7ec3d5838b41ff44073
2 / The unsuccessful fix
Exit 1"""Failure Map reference implementation. Python standard library only."""
import json
import math
N = 1
observations = []
def solve(sun_az, sun_el, mirror, receiver):
if sun_el <= 0:
return None
dx = [receiver[i] - mirror[i] for i in range(3)]
d = math.sqrt(sum(v * v for v in dx))
if d == 0:
return 'invalid'
t = [v / d for v in dx]
az = math.radians(sun_az)
el = math.radians(sun_el)
s = [math.sin(az), math.cos(az), math.sin(el)]
h = [s[i] + t[i] for i in range(3)]
m = math.sqrt(sum(v * v for v in h))
h = [v / m for v in h]
n_az = math.degrees(math.atan2(h[0], h[1])) % 360
n_el = math.degrees(math.asin(h[2]))
return [round(n_az, 3), round(n_el, 3)]
def check(label, actual, expected):
observations.append({"check": label, "actual": actual, "expected": expected, "passed": actual == expected})
fixtures = [[['boundary: receiver at mirror', [180, 40, [0, 0, 0], [0, 0, 0]], 'invalid'],
['normal: receiver due north on tower', [180, 40, [0, -40, 0], [0, 0, 40]], [180.0, 87.5]],
['boundary: night', [90, 0, [10, 10, 0], [0, 0, 50]], None],
['regression: sun vector components', [270, 60, [6, -1, 1], [0, 5, 0]], [300.29, 28.278]],
['regression: sun vector components (partial repair)', [270, 80, [-16, -19, 1], [0, 0, 0]],
[31.588, 46.473]],
['control 1', [90, -3, [-12, 43, 1], [0, 0, 0]], None],
['control 2', [200, 60, [-10, 12, 0], [0, 0, 60]], [180.839, 70.11]]],
[['boundary: receiver at mirror', [180, 40, [0, 0, 0], [0, 0, 0]], 'invalid'],
['normal: receiver due north on tower', [180, 40, [0, -40, 0], [0, 0, 40]], [180.0, 87.5]],
['boundary: night', [90, 0, [10, 10, 0], [0, 0, 50]], None],
['regression: sun vector components', [315, 25, [47, -28, 0], [0, 0, 60]], [308.943, 36.527]],
['regression: sun vector components (partial repair)', [90, 25, [-19, 40, 2], [-10, 5, 60]],
[116.262, 47.684]],
['control 1', [135, 60, [31, 24, 2], [10, 0, 60]], [177.1, 67.63]],
['control 2', [225, -3, [-22, -29, 0], [0, 5, 30]], None]],
[['boundary: receiver at mirror', [180, 40, [0, 0, 0], [0, 0, 0]], 'invalid'],
['normal: receiver due north on tower', [180, 40, [0, -40, 0], [0, 0, 40]], [180.0, 87.5]],
['boundary: night', [90, 0, [10, 10, 0], [0, 0, 50]], None],
['regression: sun vector components', [60, 40, [7, -5, 0], [0, 0, 60]], [49.649, 66.233]],
['regression: sun vector components (partial repair)', [315, 10, [-37, 26, 0], [0, -5, 60]],
[323.691, 69.061]],
['control 1', [90, 10, [-6, -40, 0], [-10, 0, 30]], [48.617, 32.61]],
['control 2', [30, 10, [-44, 5, 1], [-10, 0, 0]], [64.473, 5.033]]],
[['boundary: receiver at mirror', [180, 40, [0, 0, 0], [0, 0, 0]], 'invalid'],
['normal: receiver due north on tower', [180, 40, [0, -40, 0], [0, 0, 40]], [180.0, 87.5]],
['boundary: night', [90, 0, [10, 10, 0], [0, 0, 50]], None],
['regression: sun vector components', [315, 60, [-28, -33, 2], [-10, 5, 60]], [353.399, 62.028]],
['regression: sun vector components (partial repair)', [200, 80, [9, 42, 3], [-10, 0, 30]],
[203.614, 55.19]],
['control 1', [150, 40, [16, -12, 0], [0, 0, 0]], [261.353, 56.729]],
['control 2', [90, 0, [46, -4, 0], [0, -5, 0]], None]],
[['boundary: receiver at mirror', [180, 40, [0, 0, 0], [0, 0, 0]], 'invalid'],
['normal: receiver due north on tower', [180, 40, [0, -40, 0], [0, 0, 40]], [180.0, 87.5]],
['boundary: night', [90, 0, [10, 10, 0], [0, 0, 50]], None],
['regression: sun vector components', [200, 10, [41, -47, 2], [0, 0, 30]], [255.744, 31.113]],
['regression: sun vector components (partial repair)', [225, 40, [-5, 13, 2], [-10, -5, 30]],
[212.668, 49.07]],
['control 1', [250, -3, [-49, 36, 3], [-10, 5, 60]], None],
['control 2', [300, 80, [-27, -17, 1], [-10, 0, 60]], [18.225, 78.974]]]]
for label, args, expected in fixtures[N-1]:
check(label, solve(*args), expected)
print(json.dumps({"observations": observations, "passed": all(x["passed"] for x in observations)}, ensure_ascii=False))
raise SystemExit(0 if all(x["passed"] for x in observations) else 1)
| Boundary fixture | Actual | Expected | Outcome |
|---|---|---|---|
| boundary: receiver at mirror | invalid | invalid | Passed |
| normal: receiver due north on tower | [180.0, 77.758] | [180.0, 87.5] | Failed |
| boundary: night | None | None | Passed |
| regression: sun vector components | [292.418, 22.134] | [300.29, 28.278] | Failed |
| regression: sun vector components (partial repair) | [335.001, 48.242] | [31.588, 46.473] | Failed |
| control 1 | None | None | Passed |
| control 2 | [189.061, 57.962] | [180.839, 70.11] | Failed |
SHA-256 / 378bb088800e68f5ce70b899cef094073fc0c75bfd6b1ef021ec71b818169bd0
3 / The verified repair
Exit 0"""Failure Map reference implementation. Python standard library only."""
import json
import math
N = 1
observations = []
def solve(sun_az, sun_el, mirror, receiver):
if sun_el <= 0:
return None
dx = [receiver[i] - mirror[i] for i in range(3)]
d = math.sqrt(sum(v * v for v in dx))
if d == 0:
return 'invalid'
t = [v / d for v in dx]
az = math.radians(sun_az)
el = math.radians(sun_el)
s = [math.cos(el) * math.sin(az), math.cos(el) * math.cos(az), math.sin(el)]
h = [s[i] + t[i] for i in range(3)]
m = math.sqrt(sum(v * v for v in h))
h = [v / m for v in h]
n_az = math.degrees(math.atan2(h[0], h[1])) % 360
n_el = math.degrees(math.asin(h[2]))
return [round(n_az, 3), round(n_el, 3)]
def check(label, actual, expected):
observations.append({"check": label, "actual": actual, "expected": expected, "passed": actual == expected})
fixtures = [[['boundary: receiver at mirror', [180, 40, [0, 0, 0], [0, 0, 0]], 'invalid'],
['normal: receiver due north on tower', [180, 40, [0, -40, 0], [0, 0, 40]], [180.0, 87.5]],
['boundary: night', [90, 0, [10, 10, 0], [0, 0, 50]], None],
['regression: sun vector components', [270, 60, [6, -1, 1], [0, 5, 0]], [300.29, 28.278]],
['regression: sun vector components (partial repair)', [270, 80, [-16, -19, 1], [0, 0, 0]],
[31.588, 46.473]],
['control 1', [90, -3, [-12, 43, 1], [0, 0, 0]], None],
['control 2', [200, 60, [-10, 12, 0], [0, 0, 60]], [180.839, 70.11]]],
[['boundary: receiver at mirror', [180, 40, [0, 0, 0], [0, 0, 0]], 'invalid'],
['normal: receiver due north on tower', [180, 40, [0, -40, 0], [0, 0, 40]], [180.0, 87.5]],
['boundary: night', [90, 0, [10, 10, 0], [0, 0, 50]], None],
['regression: sun vector components', [315, 25, [47, -28, 0], [0, 0, 60]], [308.943, 36.527]],
['regression: sun vector components (partial repair)', [90, 25, [-19, 40, 2], [-10, 5, 60]],
[116.262, 47.684]],
['control 1', [135, 60, [31, 24, 2], [10, 0, 60]], [177.1, 67.63]],
['control 2', [225, -3, [-22, -29, 0], [0, 5, 30]], None]],
[['boundary: receiver at mirror', [180, 40, [0, 0, 0], [0, 0, 0]], 'invalid'],
['normal: receiver due north on tower', [180, 40, [0, -40, 0], [0, 0, 40]], [180.0, 87.5]],
['boundary: night', [90, 0, [10, 10, 0], [0, 0, 50]], None],
['regression: sun vector components', [60, 40, [7, -5, 0], [0, 0, 60]], [49.649, 66.233]],
['regression: sun vector components (partial repair)', [315, 10, [-37, 26, 0], [0, -5, 60]],
[323.691, 69.061]],
['control 1', [90, 10, [-6, -40, 0], [-10, 0, 30]], [48.617, 32.61]],
['control 2', [30, 10, [-44, 5, 1], [-10, 0, 0]], [64.473, 5.033]]],
[['boundary: receiver at mirror', [180, 40, [0, 0, 0], [0, 0, 0]], 'invalid'],
['normal: receiver due north on tower', [180, 40, [0, -40, 0], [0, 0, 40]], [180.0, 87.5]],
['boundary: night', [90, 0, [10, 10, 0], [0, 0, 50]], None],
['regression: sun vector components', [315, 60, [-28, -33, 2], [-10, 5, 60]], [353.399, 62.028]],
['regression: sun vector components (partial repair)', [200, 80, [9, 42, 3], [-10, 0, 30]],
[203.614, 55.19]],
['control 1', [150, 40, [16, -12, 0], [0, 0, 0]], [261.353, 56.729]],
['control 2', [90, 0, [46, -4, 0], [0, -5, 0]], None]],
[['boundary: receiver at mirror', [180, 40, [0, 0, 0], [0, 0, 0]], 'invalid'],
['normal: receiver due north on tower', [180, 40, [0, -40, 0], [0, 0, 40]], [180.0, 87.5]],
['boundary: night', [90, 0, [10, 10, 0], [0, 0, 50]], None],
['regression: sun vector components', [200, 10, [41, -47, 2], [0, 0, 30]], [255.744, 31.113]],
['regression: sun vector components (partial repair)', [225, 40, [-5, 13, 2], [-10, -5, 30]],
[212.668, 49.07]],
['control 1', [250, -3, [-49, 36, 3], [-10, 5, 60]], None],
['control 2', [300, 80, [-27, -17, 1], [-10, 0, 60]], [18.225, 78.974]]]]
for label, args, expected in fixtures[N-1]:
check(label, solve(*args), expected)
print(json.dumps({"observations": observations, "passed": all(x["passed"] for x in observations)}, ensure_ascii=False))
raise SystemExit(0 if all(x["passed"] for x in observations) else 1)
| Boundary fixture | Actual | Expected | Outcome |
|---|---|---|---|
| boundary: receiver at mirror | invalid | invalid | Passed |
| normal: receiver due north on tower | [180.0, 87.5] | [180.0, 87.5] | Passed |
| boundary: night | None | None | Passed |
| regression: sun vector components | [300.29, 28.278] | [300.29, 28.278] | Passed |
| regression: sun vector components (partial repair) | [31.588, 46.473] | [31.588, 46.473] | Passed |
| control 1 | None | None | Passed |
| control 2 | [180.839, 70.11] | [180.839, 70.11] | Passed |
SHA-256 / 8a6a146980872473893733b5897057bf10ed1243d31832fac4dc204e9566ff5c
Verification & scope
Deterministic stipulated toy contract for teaching; no claim of conformance with any standard, vendor protocol or production controller. This reproducer isolates one failure mechanism. Results cover the supplied fixtures. Variants within a family share a test contract and should remain grouped when constructing evaluation splits. Related mechanisms with a shared evaluation_group must also remain together; these controlled models are not independent production incidents.
Observations recorded using Python 3.12.14 at 2026-09-29T14:51:55.763455+00:00.
Case digest / cad275725f1c2e25b31316b718a325fa2314142cf4ac50f22230d28c63083108